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What family of sinusoidal functions has chaotic, unbounded, non-monotonic amplitudes and wavelengths where neither become neither infinitesimal nor infinite as x approaches positive infinity?

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  • $\begingroup$ Conway base 13 $\endgroup$ Mar 27, 2016 at 2:02
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    $\begingroup$ What is the average of a function? Are your functions integrable? Over what domain are they defined? Please clarify what you need. $\endgroup$
    – mathguy
    Mar 27, 2016 at 2:23
  • $\begingroup$ $f(x) = x\sin x$ $\endgroup$ Mar 27, 2016 at 3:30

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How about $ \big(x + \sin x \big) \cos (\pi x) $? It seems to satisfy all the conditions you've given so far. enter image description here


Amendment:

You can play this game all you want, and I'm going to stop soon, but here's a result that seems to be everything you're hoping for. It combines my previous answer and @TreeHouse196's. I'm using mathematica/wolfram alpha for the plots, and I recommend you do too. Wolfram Alpha is a popular free online tool.

enter image description here

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    $\begingroup$ Sure, the amplitude envelope here goes like "a noisy version of $x$". As you said, you can divide by $x$, and then multiply by whatever function you want to modify the amplitude envelope. Are you graphing over various ranges for $x$ in Desmos? $\endgroup$ Mar 27, 2016 at 3:52
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    $\begingroup$ The chaotic amplitude and wavelength (it does have this if you view it on the proper scale) is to mix periodic functions with rational and irrational periods. As a soft explanation, if you only combine periodic functions of rational period, the result should have a period equal to the least common multiple of the periods of the constituent functions. By picking constituent periods without a least common multiple, you avoid this and create a "chaotic wavelength". The random amplitude follows from chaotic constructive/destructive interference. $\endgroup$ Mar 27, 2016 at 4:33
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On what domain? $x^3-x$ is neither monotonic nor bounded on $\mathbb R$.

How about $f(x)=0$ for all $x$ irrational and for $x=0$, and $f(x)= \dfrac 1x$ otherwise? Not sinusoidal, average zero (zero almost everywhere), not monotonic over any interval?

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  • $\begingroup$ Don't tell me in a comment, please edit your question and tell us all what the problem is. How do you want us to guess? $\endgroup$
    – mathguy
    Mar 27, 2016 at 2:14
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What about the function $f(x) = x \cdot \cos(x)$? Here is its graph (from Wolfram Alpha):

enter image description here


The function $f(x)=x\cdot\left[\sin(x)+\sin(2x)\right]\cdot\cos(2x)$ seems to fit your criteria.

Here is a graph: enter image description here

If we graph $\vert f(x) \vert$, we can see that the absolute values of the extrema are not monotonic:

enter image description here

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I think $f(n)={(-1)}^{n}n$, over non negative integers is good example of an unbounded function which is not monotonic. sorry sir,it was a typo...

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    $\begingroup$ Uhm... It is monotonic. $\endgroup$
    – mathguy
    Mar 27, 2016 at 2:12

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